These are the exact kinds of things that a human engineer would almost certainly never do.
This kind of thing is the problem of going all in on the idea of the AI Software Factory.
It's very possible that nobody decided this at all and that's it just something an agent threw in on its own because some Chain of Thought imagined it was a clever way to correlate queries back with user accounts or something.
I'm not sure what the issue is with publishing your email address in principle, you are reaching out, you want to inquire about some business. If anything use a different address. But why would you asymmetrically expect businesses with websites to bend over to give you all the advantages, they need some assurances as well.
Anonimity usually works one way, one party can be anonymous, not both, if you come from consumer land, it's the companies exposing themselves and doing all the compliance, if you choose privacy and force the companies to expose themselves, you'll get your wish granted, but there will be a cost, it's unlikely you'll be in the more advantageous position.
A more sane implementation of this would involve a prompt the first time an agent attempts to connect to each of the sites that Anthropic thinks it necessary to send this email header to, asking if this is OK (and bailing on all requests to that domain if the prompt was rejected).
I'm sure the answer will be to update AGENTS.md to not do this particular thing, expose email in this way, which isn't the same as having sense.